Mechanical Energy Balance for Laminar Flow

When the flow is slow and laminar, we use a different non-dimensionalization for the pressure. Divide the pressure by the viscosity times the velocity divided by a characteristic length  

We then multiply by p xjrj to get another non-dimensional form of the mechanical energy balance  

As noted above, for a fully developed flow through a tube with a constant diameter, d, the kinetic energy does not change from one end to the other. As we, again, already have a formula for the pressure drop in terms of the friction factor we find  

the viscous dissipation term in Eq. (11) for laminar flow in a straight channel is  

Both Eqs. (10) and (11) express the same physics. Equation (10) is more useful at high velocity in turbulent flow when /is a slowly varying function of Reynolds number, and Eq. (11) is more useful at low velocity in laminar flow. The non-dimensional pressures in these two equations are different, but they are related  

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